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PPrroobbaabbiilliittyy 
DDiissttrriibbuuttiioonnss
RRaannddoomm VVaarriiaabbllee 
• Random variable 
– Outcomes of an experiment 
expressed numerically 
– e.g.: Throw a die twice; Count the 
number of times the number 6 
appears (0, 1 or 2 times)
DDiissccrreettee RRaannddoomm VVaarriiaabbllee
DDiissccrreettee PPrroobbaabbiilliittyy 
DDiissttrriibbuuttiioonn EExxaammppllee 
Event: Toss 2 Coins. Count # Tails. 
Probability Distribution 
Values Probability 
0 1/4 = .25 
1 2/4 = .50 
2 1/4 = .25 
T 
T 
T T
Discrete Probability 
Distributions 
Binomial Poisson
BBiinnoommiiaall DDiissttrriibbuuttiioonn- AAssssuummppttiioonnss 
• “n” Identical & finite trials 
– e.g.: 15 tosses of a coin; 10 light bulbs taken from a 
warehouse 
• Two mutually exclusive outcomes on each 
trial 
– e.g.: Heads or tails in each toss of a coin; defective or 
not defective light bulb 
• Trials are independent of each other 
– The outcome of one trial does not affect the outcome of 
the other. 
• Constant probability for each trial 
– e.g.: Probability of getting a tail is the same each time a 
coin is tossed
BBiinnoommiiaall DDiissttrriibbuuttiioonn 
• A random variable X is said to follow 
Binomial distribution if it assumes only 
non negative values and its function is 
given by 
P x n ! 
- - 
( ) ( ) x ( ) n x 
= p 1 
p 
X x n - 
x 
! ! 
• x: No. of “successes” in a sample, 
x=0,1,2..,n. 
• n: Number of trials. 
• p: probability of each success. 
• q: probability of each failure (q=1-p) 
•Notation: X ~ B (n, p).
BBiinnoommiiaall 
DDiissttrriibbuuttiioonn CChhaarraacctteerriissttiiccss 
• Mean 
EX: 
m = E ( X ) = np 
m = np = 5( .1) = .5 
• Variance and 
standard deviation- 
Ex: 
n = 5 p = 0.1 
.6 
.4 
.2 
0 
0 1 2 3 4 5 
X 
P(X) 
( ) 
( ) 
2 1 
np p 
np 1 
p 
s = np(1- p) = 5( .1) (1-.1) = .6708 
s 
s 
= - 
= -
EExxaammppllee 
• The experiment: Randomly draw red ball with 
replacement from an urn containing 10 red balls 
and 20 black balls. 
• Use S to denote the outcome of drawing a red 
ball and F to denote the outcome of a black ball. 
• Then this is a binomial experiment with p =1/3. 
• Q: Would it still be a binomial experiment if the 
balls were drawn without replacement?
QQuueessttiioonnss ffoorr pprraaccttiiccee 
• Ten coins are thrown simultaneously. Find the 
probability of getting at least seven heads. 
• A & B play a game in which their chances of winning are 
in the ratio 3:2. find A’s chance of winning at least 3 
games out of the 5 games played. 
• Mr. Gupta applies for a personal loan of Rs 150,000 
from a nationalized bank to repair his house. The loan 
offer informed him that over the years bank has 
received about 2920 loan applications per year and that 
the prob. Of approval was on average, about 0.85. Mr. 
Gupta wants to know the average and standard 
deviation of the number of loans approved per year.
• The incidence of a certain disease in 
such that on the average 20% of workers 
suffer from it. If 10 workers are 
selected at random, find the probability 
that 
1. Exactly 2 workers suffer from the 
disease. 
2. Not more than 2 workers suffer from 
the disease. 
• Bring out the fallacy, if any 
1. The mean of a binomial distribution is 15 
and its standard deviation is 5. 
2. Find the binomial distribution whose 
mean is 6 and variance is 4.
• The probability that an evening college student 
will be graduate is 0.4.Determine the probability 
that out of 5 students 
• None will be graduate. 
• One will be graduate. 
• At least one will be graduate. 
• Multiple choice test consists of 8 questions and 
three answers to each question (of which only 
one is correct). A student answers each question 
by rolling a balanced die and marking the first 
answer if he gets 1 or 2, the second answer if he 
gets 3 or 4 and the third answer if he gets 5 or 
6. To get a distinction, student must secure at 
least 75% correct answers.If there is no 
negative marking, what is the probability that 
student secures a distinction?
Q. Assuming that half the population are 
Consumers of rice and assuming that 100 
investigators can take sample of 10 
individuals to see whether they are rice 
consumers , how many investigations 
would you expect to report that three 
people or less were consumers of rice? 
Q. Five coins are tossed 3200 times. Find 
the frequencies of the distribution of 
heads and tails and tabulate the result. 
Also calculate the mean number of 
successes and S.D.
CCrreeddiitt CCaarrdd EExxaammppllee 
• Records show that 5% of the customers in a 
shoe store make their payments using a credit 
card. 
• This morning 8 customers purchased shoes. 
• Use the binomial table to answer the following 
questions. 
1. Find the probability that exactly 6 customers 
did not use a credit card. 
2. What is the probability that at least 3 
customers used a credit card?
CCrreeddiitt CCaarrdd EExxaammppllee 
3. What is the expected number of customers 
who used a credit card? 
4. What is the standard deviation of the 
number of customers who used a credit 
card?
PPaarrkkiinngg EExxaammppllee 
• Sarah drives to work everyday, but does not own a parking 
permit. She decides to take her chances and risk getting a 
parking ticket each day. Suppose 
– A parking permit for a week (5 days) cost $ 30. 
– A parking fine costs $ 50. 
– The probability of getting a parking ticket each day is 
0.1. 
– Her chances of getting a ticket each day is independent 
of other days. 
– She can get only 1 ticket per day. 
• What is her probability of getting at least 1 parking ticket 
in one week (5 days)? 
• What is the expected number of parking tickets that Sarah 
will get per week? 
• Is she better off paying the parking permit in the long run?
7. binomial distribution
BBiinnoommiiaall DDiissttrriibbuuttiioonn- 
DDiiffffeerreenntt ssiittuuaattiioonnss 
Random experiments and random variable
SSiittuuaattiioonnss CCoonnttdd.... 
Random experiments and random variables
PPooiissssoonn 
DDiissttrriibbuuttiioonn
AAssssuummppttiioonnss 
Applicable when- 
1) No. of trials is indefinitely large n®¥ 
2) Probability of success p for each trial 
is very small. p ®0 
3) Mean is a finite number given by 
np = l
PPooiissssoonn DDiissttrriibbuuttiioonn FFuunnccttiioonn 
X P ( X 
) 
e X 
! 
( ) 
P X X 
X 
: probability of "successes" given 
: number of "successes" per unit 
: expected (average) number of "successes" 
: 2.71828 (base of natural logs) 
e 
ll 
l 
l 
- 
= X =0,1,2… 
X ~ P(l)
PPooiissssoonn DDiissttrriibbuuttiioonn 
CChhaarraacctteerriissttiiccss 
( ) 
m l 
= = 
=å 
( ) 
E X 
N 
1 
i i 
i 
X P X 
= 
• Mean 
• Variance & Standard deviation 
s 2 =l s = l
Example 
Arrivals at a bus-stop follow a 
Poisson distribution with an 
average of 4.5 every quarter of 
an hour. 
(assume a maximum of 20 
arrivals in a quarter of an hour) 
and calculate the probability of 
fewer than 3 arrivals in a 
quarter of an hour.
The probabilities of 0 up to 2 arrivals can 
be calculated directly from the formula 
-ll 
x p () 
x 
e ! 
x 
= 
4.504.5 (0) 
0! 
p e 
- 
= 
with l =4.5 
So p(0) = 0.01111
Similarly p(1)=0.04999 and p(2)=0.11248 
So the probability of fewer than 3 arrivals is 
0.01111+ 0.04999 + 0.11248 =0.17358
QQuueessttiioonnss ffoorr pprraaccttiiccee 
• Suppose on an average 1 house in 1000 in a 
certain district has a fire during a year.If 
there are 2000 houses in that district,what is 
the probability that exactly 5 houses will have 
a fire during the year? 
• If 3% of the electric bulbs manufactured by 
a company are defective, find the probability 
that in a sample of 100 bulbs 
a) 0 b) 5 bulbs c) more than 5 
d) between 1 and 3 
e) less than or equal to 2 bulbs are defective
• Comment on the following for a Poisson 
distribution with Mean = 3 and s.D = 2 
• In a Poisson distribution if p(2)= 2/3 p(1). 
Find 
i) mean ii) standard deviation 
iii)P(3) iv) p(x > 3) 
• A car hire firm has two cars, which it hires 
out day by day.The number of demands for a 
car on each day is distributed as a Poisson 
distribution with mean 1.5. Calculate the 
proportion of days on which 
i) Neither car is used. 
ii) Some demand is refused.
• A manufacturer who produces medicine 
bottles,finds that 0.1% of the bottles are 
defective. The bottles are packed in boxes 
containing 500 bottles.A drug manufacturer 
buys 100 boxes from the producer of 
bottles.Find how many boxes will contain 
i) no defectives 
ii) at least two defectives 
• The following table gives the number of days 
in a 50 day period during which automobile 
accidents occurred in a city- 
No. of accidents: 0 1 2 3 4 
No. of days : 21 18 7 3 1 
Fit a Poisson distribution to data.
Note: 
In most cases, the actual number of trials is 
not known, only the average chance of 
occurrence based on the past experience 
can help in constructing the whole distribution . 
For ex: The number of accidents in any 
Particular period of time : the distribution 
is based on the mean value of occurrence of 
an event , which may also be obtained based 
on Past Experience .
• In a town 10 accidents took place in a 
span of 50 days. Assuming that the 
number of accidents per day follow 
Poisson distribution, find the probability 
that there will be 3 or more accidents in 
a day. 
HINT: 
l = 10/50 = 0.2 
Average no. of accidents per day
• Case-Let 
• An insurance company insures 4000 
people against loss of both eyes in a car 
accident.Based on previous data,the 
rates were computed on the assumption 
that on the average 10 persons in 
1,00,000 will have car accident each 
year that result in this type of 
injury.What is the probability that more 
than 4 of the insured will collect on a 
policy in a given year?
PPooiissssoonn ttyyppee SSiittuuaattiioonnss 
• Number of deaths from a disease. 
• No. of suicides reported in a city. 
• No. of printing mistakes at each page of a 
book. 
• Emission of radioactive particles. 
• No. of telephone calls per minute at a small 
business. 
• No. of paint spots per new automobile. 
• No. of arrivals at a toll booth 
• No. of flaws per bolt of cloth

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7. binomial distribution

  • 2. RRaannddoomm VVaarriiaabbllee • Random variable – Outcomes of an experiment expressed numerically – e.g.: Throw a die twice; Count the number of times the number 6 appears (0, 1 or 2 times)
  • 4. DDiissccrreettee PPrroobbaabbiilliittyy DDiissttrriibbuuttiioonn EExxaammppllee Event: Toss 2 Coins. Count # Tails. Probability Distribution Values Probability 0 1/4 = .25 1 2/4 = .50 2 1/4 = .25 T T T T
  • 6. BBiinnoommiiaall DDiissttrriibbuuttiioonn- AAssssuummppttiioonnss • “n” Identical & finite trials – e.g.: 15 tosses of a coin; 10 light bulbs taken from a warehouse • Two mutually exclusive outcomes on each trial – e.g.: Heads or tails in each toss of a coin; defective or not defective light bulb • Trials are independent of each other – The outcome of one trial does not affect the outcome of the other. • Constant probability for each trial – e.g.: Probability of getting a tail is the same each time a coin is tossed
  • 7. BBiinnoommiiaall DDiissttrriibbuuttiioonn • A random variable X is said to follow Binomial distribution if it assumes only non negative values and its function is given by P x n ! - - ( ) ( ) x ( ) n x = p 1 p X x n - x ! ! • x: No. of “successes” in a sample, x=0,1,2..,n. • n: Number of trials. • p: probability of each success. • q: probability of each failure (q=1-p) •Notation: X ~ B (n, p).
  • 8. BBiinnoommiiaall DDiissttrriibbuuttiioonn CChhaarraacctteerriissttiiccss • Mean EX: m = E ( X ) = np m = np = 5( .1) = .5 • Variance and standard deviation- Ex: n = 5 p = 0.1 .6 .4 .2 0 0 1 2 3 4 5 X P(X) ( ) ( ) 2 1 np p np 1 p s = np(1- p) = 5( .1) (1-.1) = .6708 s s = - = -
  • 9. EExxaammppllee • The experiment: Randomly draw red ball with replacement from an urn containing 10 red balls and 20 black balls. • Use S to denote the outcome of drawing a red ball and F to denote the outcome of a black ball. • Then this is a binomial experiment with p =1/3. • Q: Would it still be a binomial experiment if the balls were drawn without replacement?
  • 10. QQuueessttiioonnss ffoorr pprraaccttiiccee • Ten coins are thrown simultaneously. Find the probability of getting at least seven heads. • A & B play a game in which their chances of winning are in the ratio 3:2. find A’s chance of winning at least 3 games out of the 5 games played. • Mr. Gupta applies for a personal loan of Rs 150,000 from a nationalized bank to repair his house. The loan offer informed him that over the years bank has received about 2920 loan applications per year and that the prob. Of approval was on average, about 0.85. Mr. Gupta wants to know the average and standard deviation of the number of loans approved per year.
  • 11. • The incidence of a certain disease in such that on the average 20% of workers suffer from it. If 10 workers are selected at random, find the probability that 1. Exactly 2 workers suffer from the disease. 2. Not more than 2 workers suffer from the disease. • Bring out the fallacy, if any 1. The mean of a binomial distribution is 15 and its standard deviation is 5. 2. Find the binomial distribution whose mean is 6 and variance is 4.
  • 12. • The probability that an evening college student will be graduate is 0.4.Determine the probability that out of 5 students • None will be graduate. • One will be graduate. • At least one will be graduate. • Multiple choice test consists of 8 questions and three answers to each question (of which only one is correct). A student answers each question by rolling a balanced die and marking the first answer if he gets 1 or 2, the second answer if he gets 3 or 4 and the third answer if he gets 5 or 6. To get a distinction, student must secure at least 75% correct answers.If there is no negative marking, what is the probability that student secures a distinction?
  • 13. Q. Assuming that half the population are Consumers of rice and assuming that 100 investigators can take sample of 10 individuals to see whether they are rice consumers , how many investigations would you expect to report that three people or less were consumers of rice? Q. Five coins are tossed 3200 times. Find the frequencies of the distribution of heads and tails and tabulate the result. Also calculate the mean number of successes and S.D.
  • 14. CCrreeddiitt CCaarrdd EExxaammppllee • Records show that 5% of the customers in a shoe store make their payments using a credit card. • This morning 8 customers purchased shoes. • Use the binomial table to answer the following questions. 1. Find the probability that exactly 6 customers did not use a credit card. 2. What is the probability that at least 3 customers used a credit card?
  • 15. CCrreeddiitt CCaarrdd EExxaammppllee 3. What is the expected number of customers who used a credit card? 4. What is the standard deviation of the number of customers who used a credit card?
  • 16. PPaarrkkiinngg EExxaammppllee • Sarah drives to work everyday, but does not own a parking permit. She decides to take her chances and risk getting a parking ticket each day. Suppose – A parking permit for a week (5 days) cost $ 30. – A parking fine costs $ 50. – The probability of getting a parking ticket each day is 0.1. – Her chances of getting a ticket each day is independent of other days. – She can get only 1 ticket per day. • What is her probability of getting at least 1 parking ticket in one week (5 days)? • What is the expected number of parking tickets that Sarah will get per week? • Is she better off paying the parking permit in the long run?
  • 18. BBiinnoommiiaall DDiissttrriibbuuttiioonn- DDiiffffeerreenntt ssiittuuaattiioonnss Random experiments and random variable
  • 19. SSiittuuaattiioonnss CCoonnttdd.... Random experiments and random variables
  • 21. AAssssuummppttiioonnss Applicable when- 1) No. of trials is indefinitely large n®¥ 2) Probability of success p for each trial is very small. p ®0 3) Mean is a finite number given by np = l
  • 22. PPooiissssoonn DDiissttrriibbuuttiioonn FFuunnccttiioonn X P ( X ) e X ! ( ) P X X X : probability of "successes" given : number of "successes" per unit : expected (average) number of "successes" : 2.71828 (base of natural logs) e ll l l - = X =0,1,2… X ~ P(l)
  • 23. PPooiissssoonn DDiissttrriibbuuttiioonn CChhaarraacctteerriissttiiccss ( ) m l = = =å ( ) E X N 1 i i i X P X = • Mean • Variance & Standard deviation s 2 =l s = l
  • 24. Example Arrivals at a bus-stop follow a Poisson distribution with an average of 4.5 every quarter of an hour. (assume a maximum of 20 arrivals in a quarter of an hour) and calculate the probability of fewer than 3 arrivals in a quarter of an hour.
  • 25. The probabilities of 0 up to 2 arrivals can be calculated directly from the formula -ll x p () x e ! x = 4.504.5 (0) 0! p e - = with l =4.5 So p(0) = 0.01111
  • 26. Similarly p(1)=0.04999 and p(2)=0.11248 So the probability of fewer than 3 arrivals is 0.01111+ 0.04999 + 0.11248 =0.17358
  • 27. QQuueessttiioonnss ffoorr pprraaccttiiccee • Suppose on an average 1 house in 1000 in a certain district has a fire during a year.If there are 2000 houses in that district,what is the probability that exactly 5 houses will have a fire during the year? • If 3% of the electric bulbs manufactured by a company are defective, find the probability that in a sample of 100 bulbs a) 0 b) 5 bulbs c) more than 5 d) between 1 and 3 e) less than or equal to 2 bulbs are defective
  • 28. • Comment on the following for a Poisson distribution with Mean = 3 and s.D = 2 • In a Poisson distribution if p(2)= 2/3 p(1). Find i) mean ii) standard deviation iii)P(3) iv) p(x > 3) • A car hire firm has two cars, which it hires out day by day.The number of demands for a car on each day is distributed as a Poisson distribution with mean 1.5. Calculate the proportion of days on which i) Neither car is used. ii) Some demand is refused.
  • 29. • A manufacturer who produces medicine bottles,finds that 0.1% of the bottles are defective. The bottles are packed in boxes containing 500 bottles.A drug manufacturer buys 100 boxes from the producer of bottles.Find how many boxes will contain i) no defectives ii) at least two defectives • The following table gives the number of days in a 50 day period during which automobile accidents occurred in a city- No. of accidents: 0 1 2 3 4 No. of days : 21 18 7 3 1 Fit a Poisson distribution to data.
  • 30. Note: In most cases, the actual number of trials is not known, only the average chance of occurrence based on the past experience can help in constructing the whole distribution . For ex: The number of accidents in any Particular period of time : the distribution is based on the mean value of occurrence of an event , which may also be obtained based on Past Experience .
  • 31. • In a town 10 accidents took place in a span of 50 days. Assuming that the number of accidents per day follow Poisson distribution, find the probability that there will be 3 or more accidents in a day. HINT: l = 10/50 = 0.2 Average no. of accidents per day
  • 32. • Case-Let • An insurance company insures 4000 people against loss of both eyes in a car accident.Based on previous data,the rates were computed on the assumption that on the average 10 persons in 1,00,000 will have car accident each year that result in this type of injury.What is the probability that more than 4 of the insured will collect on a policy in a given year?
  • 33. PPooiissssoonn ttyyppee SSiittuuaattiioonnss • Number of deaths from a disease. • No. of suicides reported in a city. • No. of printing mistakes at each page of a book. • Emission of radioactive particles. • No. of telephone calls per minute at a small business. • No. of paint spots per new automobile. • No. of arrivals at a toll booth • No. of flaws per bolt of cloth